MatheMUSEments

February 17, 2008

Take a look at a map of the United States and locate Colorado and Wyoming. On many maps, these states look like perfect rectangles.

The laws that created these two states specify that each one lies between certain lines of latitude and longitude. (Latitude lines are drawn side-to-side on a globe; longitude lines go top-to-bottom.) Wyoming stretches from 41°N to 45°N latitude and from 104°W to 111°W longitude. Colorado's borders, meanwhile, are defined by 37°N and 41°N latitude, and 102°W to 109°W longitude. If Earth were flat, both states would be rectangles with parallel opposite sides.

But Earth is a sphere, and, on the surface a sphere, although lines of latitude are parallel, lines of longitude get closer together as you travel northward from the equator. So, the northern border of each state is a little shorter than its southern border. For instance, in Colorado, the difference is about 21 miles. The reality is that both states are shaped like trapezoids (four-sided figures with just one set of parallel sides), but on a curved surface.

There's another complication. When the states were created, surveyors had to go out into the wilderness to map the boundaries, using a compass and the stars, along with a few other tools. They followed the appropriate lines of latitude and longitude the best they could, marking the boundaries mile by mile.

The boundary between Utah and Colorado, for example, runs 276 miles from Four Corners (the only place in the United States where four states share a point) to the Wyoming border. Later surveys showed that surveying were made between mileposts 81 and 89 (northward from Four Corners) and between mileposts 100 and 110. These errors put kinks in what should have been a straight line. If you look at a highly detailed map of Colorado, you can see the kinks. There are similar errors along other borders that are supposed to be straight lines, including those of Wyoming.

Interestingly, once a border is defined on the ground and accepted by the government, it becomes official, even if it doesn't follow the written description.

Perhaps it's best to describe Colorado and Wyoming as polygonsgeometric figures with many straight sides, even though they are also curved over the surface of a sphere.Muse, February 2008, p. 15.

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Ivars Peterson is a freelance writer and editor. He was Director of Publications at the Mathematical Association of America from 2007 to 2014. As an award-winning mathematics writer, he previously worked at Science News for more than 25 years and served as editor of Science News Online and Science News for Kids. His books include The Mathematical Tourist, Islands of Truth, Newton's Clock, and Fragments of Infinity: A Kaleidoscope of Math and Art.